If $|x| + x + y = 10$ and $x + |y| - y = 12,$ find $x + y.$
Answer: If $x < 0,$ then $|x| = -x,$ so from the first equation, $y = 10.$  But then the second equation gives us $x = 12,$ contradiction, so $x \ge 0,$ which means $|x| = x.$

If $y > 0,$ then $|y| = y,$ so from the second equation, $x = 12.$  But the the first equation gives us $y = -14,$ contradiction, so $y \le 0,$ which means $|y| = -y.$

Thus, the given equations become $2x + y = 10$ and $x - 2y = 12.$  Solving, we find $x = \frac{32}{5}$ and $y = -\frac{14}{5},$ so $x + y = \boxed{\frac{18}{5}}.$